The Mean Value Theorem

There are several ways to say what the Mean Value Theorem means:

• It relates the slope of a secant line on a curve to the slope of a tangent line to the curve.
• It relates the average rate of change of a function to the instantaneous rate of change of a function.

Here's the precise mathematical statement. If y = f(x) is a function which is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there is a point c lying (strictly) between a and b such that

             f(b) - f(a)
             ----------- = f'(c).
                b - a

Here's how this equation is related to the descriptions I gave above.

• The expression on the left is the slope of the secant line joining (a,f(a)) to (b,f(b)). The expression on the right is the slope of the tangent line at (c,f(c)). The equation says that the slopes are the same, i.e. that the secant and tangent are parallel.
• The expression on the left is the average rate of change of the function f from a to b. The expression on the right is the instantaneous rate of change of f at c.

The name "Mean Value Theorem" comes from the relationship between an average rate of change and an instantaneous rate of change. "Mean" is a synonym for "average".

It's easy to see that the Mean Value Theorem makes sense by drawing a picture.

Draw the graph of y = f(x) on the interval [a,b]. Draw the secant line through the endpoints (a,f(a)) and (b,f(b)). Now slide the secant line till it becomes tangent to the curve, always keeping it parallel to its "original position". The point of tangency is the point c whose existence is guaranteed by the Mean Value Theorem; the theorem says that you will always be able to find such a point.

Note that there may be more than one value "c" that satisfies the conclusion of the Mean Value Theorem. In this picture, you see a situation where there are two x-values c and d where the tangent line is parallel to the original secant line.

• Understanding the hypotheses of the Mean Value Theorem.
• The proof of the Mean Value Theorem.
• Rolle's theorem
  • About Rolle's theorem
  • A proof of Rolle's theorem

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Last updated: June 13, 2005

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